To make the aioli

1. Combine all the ingredients in a medium bowl and stir until combined and smooth. Cover and refrigerate until ready to use.

To make the potatoes

1. Pour enough oil in a large, deep frying pan to reach ½-inch up the side of the pan. Insert a deep-fat thermometer and heat the oil to 220°F. Add half the potatoes and fry until just tender in the middle, about 15 minutes. With a slotted spoon remove to a baking sheet and fry the remaining potatoes. Remove the potatoes to the baking sheet, reserving the oil and set aside to cool, at least 30 minutes.

Meanwhile, make the Brava Sauce

1. Heat oil in a medium saucepan over medium heat. Add onion and season with salt and pepper. Cook until tender, about 6-7 minutes.
2. Add garlic and tomato paste and cook, stirring, 1 minute. Add diced tomatoes and paprika and bring to a simmer. Simmer, covered, over low heat about 10 minutes, or until flavors have combined.
3. Transfer mixture to a blender, add vinegar, and blend until smooth —be careful, hot liquids could splatter, cover blender with a kitchen towel before blending. Return sauce to saucepan, taste and add more salt as needed. Cover and keep warm while you finish the potatoes.

To finish the potatoes

1. Reheat the oil, this time to 350°F. Carefully add half of the potatoes to the hot oil and fry until crisp and golden brown, about 4-5 minutes. Transfer to a paper towel-lined plate to drain and fry remaining potatoes.
2. Place fried potatoes on a platter, season with salt and drizzle with Brava sauce and aioli. Serve immediately with sliced scallions, if desired.